A005097 -id:A005097 - cp's OEIS frontend

A163427 Primes p such that (p+1)^3/8+(p-1)/2 is also prime.

5, 7, 13, 19, 29, 31, 41, 53, 71, 101, 103, 109, 173, 191, 199, 223, 229, 233, 239, 257, 269, 277, 331, 383, 397, 431, 491, 569, 571, 599, 619, 631, 719, 733, 751, 757, 761, 823, 857, 859, 863, 887, 907, 937, 967, 971, 977, 1009, 1019, 1063, 1069, 1123, 1163

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Primes A000040(k) such that (A006254(k-1))^3+ A005097(k-1) is also prime.

			For p=5, (5+1)^3/8+(5-1)/2=27+2=29, prime, which adds p=5 to the sequence.
For p=7, (7+1)^3/8+(7-1)/2=67, prime, which adds p=7 to the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426.

[p: p in PrimesInInterval(3, 1200) | IsPrime((p+1)^3 div 8+(p-1) div 2)]; // Vincenzo Librandi, Apr 09 2013

f[n_]:=((p+1)/2)^3+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, p]],{n,6!}];lst
Select[Prime[Range[100]], PrimeQ[(# + 1)^3 / 8 + (# - 1) / 2]&] (* Vincenzo Librandi, Apr 09 2013 *)

(a(n)+1)^3/8+(a(n)-1)/2 = A163426(n).

Edited by R. J. Mathar, Aug 24 2009

A380550 List of numbers not of the form i + 3j + 4i*j for i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 19, 20, 21, 25, 26, 27, 30, 31, 32, 34, 37, 39, 40, 41, 44, 45, 47, 49, 52, 54, 55, 56, 59, 62, 65, 66, 67, 70, 72, 75, 76, 77, 81, 82, 84, 86, 89, 91, 94, 95, 102, 104, 107, 109, 110, 111, 115, 116, 117, 119, 121, 122, 124, 125, 129, 130, 135, 136, 140, 142, 144, 146, 147, 149

Offset: 1

Peter Bala, Jan 26 2025

This is a companion sequence to A380572. It is the complementary sequence to A380549.
Compare with A006093, numbers not of the form i + j + i*j, and A005097, numbers not of the form i + j + 2*i*j.
Apart from a(6) = 6, this sequence consists of all the positive integers N such that 4*N + 3 is either a prime or three times a prime.

			Factorization of 4*a(n) + 3 for n = 1..78:
[7,  11, 3*5, 19, 23, 3^3, 31, 3*13, 43, 47, 3*17, 59, 67, 71, 79, 83, 3*29, 103, 107, 3*37, 3*41, 127, 131, 139, 151, 3*53, 163, 167, 179, 3*61, 191, 199, 211, 3*73, 223, 227, 239, 251, 263, 3*89, 271, 283, 3*97, 3*101, 307, 311, 3*109, 331, 3*113, 347, 359, 367, 379, 383, 3*137, 419, 431, 439, 443, 3*149, 463, 467, 3*157, 479, 487, 491, 499, 503, 3*173, 523, 3*181, 547, 563, 571, 3*193, 587, 3*197, 599]

Peter Bala, 4*A380550(n) + 3 is either a prime or three times a prime except when n = 6

Cf. A005097, A006093, A380549, A380572.

L := 150: N := {seq(n, n= 1..L)}: S := {}:
for i from 1 to L do
  for j from 1 to L do
    if i + 3*j + 4*i*j <= L then S := `union`(S, {i+3*j+4*i*j}) end if
  end do;
end do:
N minus S;

A380572 Complement of A380509.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15, 17, 18, 22, 23, 24, 25, 27, 28, 32, 34, 35, 37, 39, 43, 44, 45, 48, 49, 50, 53, 57, 58, 59, 60, 62, 64, 67, 69, 70, 73, 77, 78, 79, 80, 84, 87, 88, 93, 95, 97, 98, 99, 100, 102, 104, 105, 108, 111, 112, 113, 114, 115, 122

Offset: 1

Davide Rotondo, Jan 27 2025

nonn

4*a(n) + 1 or (4*a(n) + 1)/3 is a prime number.

Compare with A380550, numbers not of the form i + 3*j + 4*i*j. See also A006093, numbers not of the form i + j + i*j and A005097, numbers not of the form i + j + 2*i*j. - Peter Bala, Jan 30 2025

Peter Bala, 4*A380572(n) + 1 is either a prime or three times a prime

Cf. A005097, A006093, A380509, A380550.

S := {}:
for n from 1 to 150 do
  if isprime(4*n+1) then S := `union`(S, {n}) fi;
  if type((4*n+1)*(1/3), integer) then if isprime((4*n+1)*(1/3)) then S := `union`(S, {n}) fi; fi;
end do:
S; # Peter Bala, Jan 30 2025

A053662 Numbers k such that 2k+1 divides k!+1.

Original entry on oeis.org

3, 5, 9, 21, 23, 33, 39, 51, 63, 65, 81, 89, 95, 99, 113, 131, 173, 183, 191, 209, 215, 221, 239, 245, 251, 261, 281, 285, 299, 303, 309, 315, 341, 345, 363, 369, 371, 393, 411, 419, 431, 443, 473, 495, 509, 525, 543, 545, 561, 575, 593, 645, 659, 683, 711

Offset: 1

Chris K. Caldwell, Feb 16 2000

k+1 divides k!+1 gives primes-1 by Wilson's Theorem. For the present sequence, there are 309 terms below 5000, compared with 669 primes (309/669 = 0.461...). There are 553 terms below 10000, compared with 1229 primes (553/1229 = 0.449...). - Ed Pegg Jr, Dec 05 2001

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005097, A006093, A053663.

Filtered([1..1000], n-> (Factorial(n)+1) mod (2*n+1)=0) # G. C. Greubel, May 18 2019

[n: n in [1..1000] | (Factorial(n)+1) mod (2*n+1) eq 0 ]; // G. C. Greubel, May 18 2019

A053662:=n->`if`(n!+1 mod (2*n+1) = 0, n, NULL): seq(A053662(n), n=1..1000); # Wesley Ivan Hurt, Dec 01 2015

Drop[Union[Table[If[IntegerQ[(n!+1)/(2n+1)], n], {n, 1, 1000}]], -1] (* Ed Pegg Jr, Dec 05 2001 *)
Select[Range[1000], Mod[#! +1, 2*# +1] == 0 &] (* G. C. Greubel, May 18 2019 *)

for(n=1,10^3, if((n!+1)%(2*n+1)==0, print1(n,", ")) ) \\ G. C. Greubel, May 18 2019

[n for n in (1..1000) if Mod(factorial(n)+1, 2*n+1)==0 ] # G. C. Greubel, May 18 2019

a(n) >> n log n. - Charles R Greathouse IV, Apr 16 2024

A061066 a(n) = (prime(n)^2 - 1)/8.

Original entry on oeis.org

1, 3, 6, 15, 21, 36, 45, 66, 105, 120, 171, 210, 231, 276, 351, 435, 465, 561, 630, 666, 780, 861, 990, 1176, 1275, 1326, 1431, 1485, 1596, 2016, 2145, 2346, 2415, 2775, 2850, 3081, 3321, 3486, 3741, 4005, 4095, 4560, 4656, 4851, 4950, 5565, 6216, 6441

Offset: 2

Labos Elemer, May 28 2001

nonn

This sequence is a subsequence of the triangular numbers (A000217) because (prime(n)^2-1)/8 = ((2m+1)^2-1)/8 = m(m+1)/2 where p=2m+1 for a given m. - David Morales Marciel, Oct 07 2015
The Jacobi symbol (2|p) = (-1)^((p^2-1)/8). - Michael Somos, Feb 17 2020
Number of inversions of the permutation ((2*i) mod p){1<=i<=p-1} = (2,4,...,p-1,1,3,...,p-2) of {1,2,...,p-1}, where p = prime(n). - _Jianing Song, Apr 07 2023

			a(2) = 1 because p = prime(2) = 3 and (3^2-1)/8 = 1. - _Michael Somos_, Feb 17 2020

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 307.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A000217, A005097, A024700.

[(p^2-1)/8: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

f[n_]:=(Prime[n]^2-1)/8; Array[f,66,2] (* Vladimir Joseph Stephan Orlovsky, Aug 06 2009 *)
(#^2-1)/8&/@Prime[Range[2,50]] (* Harvey P. Dale, Nov 16 2012 *)

vector(100, n, (prime(n+1)^2 - 1)/8) \\ Altug Alkan, Oct 07 2015

[(n^2-1)/8 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A000217(A005097(n-1)). - after first comment, Michel Marcus, Oct 07 2015

a(n) = (3/8)*A024700(n-2). - G. C. Greubel, May 03 2024

A061285 a(n) = 2^((prime(n) - 1)/2).

Original entry on oeis.org

2, 4, 8, 32, 64, 256, 512, 2048, 16384, 32768, 262144, 1048576, 2097152, 8388608, 67108864, 536870912, 1073741824, 8589934592, 34359738368, 68719476736, 549755813888, 2199023255552, 17592186044416, 281474976710656, 1125899906842624, 2251799813685248

Offset: 2

Labos Elemer, May 22 2001

nonn

Square root of 2^(prime(n) - 1), i.e., the smallest number that has prime(n) divisors.

Index to divisibility sequences

Cf. A000005, A000010, A005097, A005179, A006005, A006093, A034785, A217054.

Table[2^((Prime[n] - 1)/2), {n, 2, 25}] (* Amiram Eldar, Dec 23 2020 *)

a(n) = sqrt(min(x; A000005(x) = prime(n))) = sqrt(A034785(n)/2) = sqrt(2^(prime(n) - 1)) = sqrt(2^A006093(n)) = sqrt(2^phi(prime(n))) = sqrt(2^A000010(A000040(n))).

Sum_{n>=1} 1/a(n) = A217054. - Amiram Eldar, Dec 23 2020

A064673 Where the least prime p such that n = (p-1)/(q-1) and p > q is not the least prime == 1 (mod n) (A034694).

Original entry on oeis.org

24, 32, 34, 38, 62, 64, 71, 76, 80, 92, 94, 104, 110, 117, 122, 124, 129, 132, 144, 149, 152, 154, 159, 164, 167, 182, 184, 185, 188, 201, 202, 206, 212, 214, 218, 220, 225, 227, 236, 242, 244, 246, 252, 264, 269, 272, 274, 286, 290, 294

Offset: 1

Robert G. Wilson v, Oct 16 2001

			24 is in the sequence because (97-1)/(5-1) whereas the first prime ==1 (Mod 24) is 73. See the comment in A034694 about the multiplier k and it must differ from q-1 or k+1 is not prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034694, A064632, A064652. Disjoint from A005097 and A006093.

f:= proc(n) local k;
  for k from n+1 by n do
    if isprime(k) then return k fi
  od
end proc:
filter:= proc(n) local p;
    p:= f(n);
    not isprime(1+(p-1)/n)
end proc:
select(filter, [$1..1000]); # Robert Israel, May 09 2024

NextPrim[n_] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Do[p = 2; While[q = (p - 1)/n + 1; !PrimeQ[q] || q >= p, p = NextPrim[p]]; k = 1; While[ !PrimeQ[k*n + 1], k++ ]; If[p != k*n + 1, Print[n]], {n, 2, 300} ]

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

Original entry on oeis.org

3, 5, 7, 323, 11, 13, 899, 17, 19, 1763, 23, 5249, 3239, 29, 31, 979801, 5459, 37, 10763, 41, 43, 9179, 47, 9701, 10403, 53, 12319, 5646547, 59, 61, 24569, 19109, 67, 19043, 71, 73, 22499, 50819, 79, 41309, 83, 32639, 46979, 89, 34579, 39059, 125969

Offset: 1

Labos Elemer, Nov 23 2001

If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).

Donovan Johnson, Table of n, a(n) for n = 1..456

Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.

See also A005097, A047845, A014076, A001223.

max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* Jean-François Alcover, Oct 08 2016 *)

from itertools import count
from math import prod
from sympy import factorint
def A065824(n):
    for m in count(1):
        f = factorint(m)
        if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

(n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.

A067849 a(n) = max{k: f(n),...,f^k(n) are prime}, where f(m) = 2m+1 and f^k denotes composition of f with itself k times.

Original entry on oeis.org

2, 4, 1, 0, 3, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 2, 0, 0, 6, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 5, 1

Offset: 1

Joseph L. Pe, Feb 14 2002

nonn

From Glen Whitney, Sep 14 2022: (Start)
If a(n) > 3 and n > 5, then the final digit of n is 4 or 9.
a(n) > 0 if and only if n appears in A005097.
More generally, a(n) > m if and only if all of 2^k(n+1) - 1 for 0 <= k <= m are in A005097.
Creating a tile labeled by a multiple of p for a prime p with a relatively large value of a(p) is considered valuable in the game DIVE (see links). (End)

			f(2) = 5, f(f(2)) = 11, f(f(f(2))) = 23, f(f(f(f(2)))) = 47, all prime, but f^5(2) = 95 is not prime, so a(2) = 4.

Glen Whitney, Table of n, a(n) for n = 1..10000
Number-theoretic puzzle game DIVE
Glen Whitney, Python program to compute a(n)

f[n_] := Module[{a = 2n + 1, i = 0}, While[PrimeQ[a], i++; a = 2a + 1]; i]; Table[f[i], {i, 1, 60}]

a(n) = {my(nb = 0, newn); while (isprime(newn=2*n+1), nb++; n = newn); nb;} \\ Michel Marcus, Nov 10 2018

More terms from Michel Marcus, Nov 10 2018

A073409 Largest prime factor of the denominator of the Bernoulli number B(2*n) (A002445).

Original entry on oeis.org

3, 5, 7, 5, 11, 13, 3, 17, 19, 11, 23, 13, 3, 29, 31, 17, 3, 37, 3, 41, 43, 23, 47, 17, 11, 53, 19, 29, 59, 61, 3, 17, 67, 5, 71, 73, 3, 5, 79, 41, 83, 43, 3, 89, 31, 47, 3, 97, 3, 101, 103, 53, 107, 109, 23, 113, 7, 59, 3, 61, 3, 5, 127, 17, 131, 67, 3, 137, 139, 71, 3, 73, 3, 149

Offset: 1

Benoit Cloitre, Aug 23 2002

Least k such that k!*B(2n) is an integer where B(2n) denotes the 2n-th Bernoulli number.
a((p-1)/2) = p, where p is odd prime. All odd primes appear in this sequence. The very first appearance of odd prime p is a((p-1)/2). - Alexander Adamchuk, Jul 31 2006
Conjecture: a(n) is the largest prime p <= 2n+1 such that p * A000367(n) == - A002445(n) (mod p^2) for n > 0. Note that 2^(2n) == 1 (mod a(n)), since a(n) is the largest prime p such that b^(2n)== 1 (mod p) for every b coprime to p; i.e., a(n) is the largest prime p such that p-1 | 2n. - Thomas Ordowski, May 17 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000367, A002445, A005097.

Table[FactorInteger[Denominator[BernoulliB[2n]]][[ -1, 1]], {n, 100}]

a(n)=
{
    my(bd=1);
    forprime (p=2, 2*n+1, if( (2*n)%(p-1)==0, bd=p ) );
    return(bd);
}
/* Joerg Arndt, May 06 2012 */

a(n)=my(p); fordiv(n,d, if(isprime(p=2*n/d+1), return(p))) \\ Charles R Greathouse IV, Jun 08 2020

cp's OEIS Frontend

A163427 Primes p such that (p+1)^3/8+(p-1)/2 is also prime.

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A380550 List of numbers not of the form i + 3*j + 4*i*j for i, j >= 1.

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Maple

A380572 Complement of A380509.

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Maple

A053662 Numbers k such that 2k+1 divides k!+1.

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A061066 a(n) = (prime(n)^2 - 1)/8.

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A061285 a(n) = 2^((prime(n) - 1)/2).

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A064673 Where the least prime p such that n = (p-1)/(q-1) and p > q is not the least prime == 1 (mod n) (A034694).

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A380550 List of numbers not of the form i + 3j + 4i*j for i, j >= 1.

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.